We (our district’s 2 Intermediate Math Liaisons) are doing a “Getting Started with Number Talks” workshop on our ProD day this upcoming Monday. This is the last Number Talks workshop we will offer this year, and we have decided to open it to K-10 teachers. We have been getting quite a few requests for Number Talks in primary and secondary grades, so we thought we would stretch ourselves.
For part of our workshop, we are going to showcase the Number Clothesline as a nice complement for Number Talks. In the past, we have done this activity with fractions, decimals, and percentage cards (see this post), but we wanted to try to differentiate a bit for our primary and secondary attendees. I have been intrigued by the idea of using a double clothesline for algebra concepts and so for our secondary extension, we are going to try this activity from Andrew Stadel’s Estimation 180 website.
I was trying to imagine what a beginning clothesline might look like for our early primary students, and so I made up some cards with numbers, dots, ten frames and fingers (after reading this fabulous article by Jo Boaler). I thought I would do a test run this weekend with my own kiddos (Kindergarten and Grade 2).
We set up the double number line in the living room and this is how it went down…
Overall, a pretty successful test run. A few thoughts…
They both enjoyed the activity, especially the different types of pictures.
They were both disappointed that there were no numbers between 10 and 20 for the bottom number line. I can’t really do that with fingers, but maybe I will make ten frame cards up to 20 and look for domino pictures (at least up to 15’s… do dominos go up to double 20’s?).
As usual, I am impressed with how kids solve problems when they are left to themselves to figure out what makes sense. They didn’t need me to help them with anything – they figured out how to make the right spaces, what to do about missing numbers, what to do about double numbers etc. all on their own. Another good reminder that sometimes the most powerful teaching is to set the stage carefully, ask good questions and then stand back and let the kids do the thinking!!
If you would like to have these cards for your class, you can download them below. I will update this file if I manage to make some more ten frame or domino cards, but for now they only have numbers 1-10 in the dots, hands and ten frames.
We had our fourth meeting of our “Making Number Talks Matter” book club last night. Our focus for this meeting was on Fractions. We spoke a little bit about decimals and percents, but we spent most of our time looking at how we can support our students in developing conceptual understandings of fractions.
This blog post is meant to serve as a recap for those who were there, a fill-in for those who couldn’t make it, and a record for anyone else who is interested!
We started off our meeting with our usual conversation about how things are going in classrooms with number talks. Some participants shared their ideas for how they are keeping track of student thinking. One teacher has tested out incorporating our student self-assessment and another has been using her document camera instead of the whiteboard to record student thinking – she then has a record of strategies being used with student names attached to help inform her Number Talks planning. It is so inspiring to hear about how excited students are about participating in Number Talks. I hope you will all continue to carve out time in your class for them!
We then looked at a “Fractions on the Number Line” activity as a group. We used a double number line for this activity. We placed the benchmarks of 0, 1/2 and 1 on the top number line and then each participant had a number to place on the line. First, we had teachers talk in groups to order the numbers at their table and then one-by-one the tables came up to put their numbers on the bottom number line, re-arranging as necessary to make it make sense.
From this activity, we moved on to talking about the BIG IDEAS for fractional thinking in the new Grade 4-7 curriculum. We used this quote from the book as our jumping-off point:
…for success in high school, there is no avoiding fractions.
We talked about: what do our students struggle with in terms of fractional thinking? And: what do we want our students to understand about fractions?
Some thoughts that came up:
We want our students to understand that the size of the piece changes depending on the size of the whole. It is possible to have a quarter that is bigger than a half if the two wholes are different.
We want our students to understand that fractional pieces have to be the same size but not necessarily the same shape.
We want our students to understand that fractions are numbers that exist on the number line.
We want to help our students make connections between their existing understanding of number and their understanding of fractions.
We then looked at the BIG IDEAS from the curriculum from Grades 3 – 9: where are our students coming from in primary, and where do we want them to go in secondary? Now that all the fraction operations have been moved to Grade 8, we have the opportunity to solidify a conceptual understanding of fractions in elementary school so that students are prepared for fraction operations and linking of fractions to algebra in Grades 8 and 9.
From here, the teachers did another activity that connects a visual representation of a fraction to its place on the number line. (Activity adapted from this blog – printable download of the activity cards are available). Teachers coloured in a section of the given square and determined what fractional part they coloured. They then placed their fraction on the number line again.
As a wrap-up, we briefly reviewed the other three types of Number Talks for fractions that are described in the book: More or less (give a fraction and have students defend whether it is more or less than a half); Closer to 0, Closer to 1/2 or Closer to 1 (give a fraction and have students decide which benchmark it is closer to), and Which is Greater (give two fractions and have students defend which one is greater).
Last but not least, we had a mini “make and take” – teachers took home yarn for a double number line and a package with coloured fraction, decimal and percent cards. I will update this post with a link to the printable package as soon as I add some improper fractions and mixed numbers to it! I will also have these packages at our final meeting for people who were unable to join us this week.
Here are a few useful links that we talked about in our meeting:
The Teacher Studio – this blog has a fabulous series of ideas for teaching fractions conceptually. Scroll down and click on the “fractions” label on the right-hand side to find all the blog posts labelled as fractions.
Is this shape fourths? – this activity by the Teacher Studio has students defending their ideas about fractional parts. A great extension activity from your Number Talks routine.
One of my professional goals this fall as a Math Liaison in my district is to spread the message of Number Talks far and wide in intermediate classrooms in my district. Between the readings I have done (Making Number Talks Matter and Number Talks, blog posts, articles), the Pro-D workshops I have led (Number Talks book club, Number Talks and the Curricular Competencies, Intro to Number Talks), and the various Gr. 4-7 classrooms I have visited, I am starting to feel like a bit of an “expert” on the subject… and yet, Number Talks are still complicated and challenging. I think that’s one of the things I like most about the Number Talks routine – it is simple enough to be accessible, but challenging enough to keep both students and teachers engaged. So, today I thought I would share a blog post about a “failed” number talk that I have been pondering and what I learned from the experience.
I was visiting a Grade 4 class – This was my third visit to this classroom this year, and I have done Number Talks with them on each visit. We have done some dot talks and some addition number talks, and on this visit, we were going to be working on subtraction. Students in this particular class (and at this school in general) are very capable but tend to just do the traditional algorithm in their heads – this happens much more frequently here than at other schools that I visit (my theory is that it is related to high levels of parental involvement).
Rather than just one number talk, I brought a number string with me… My plan:
I really thought that these (especially the first one) were going to be easy, but when we got going on the first question, I ended up with 4 different answers. I have led a lot of number talks with multiple answers, and 99% of the time (100% of the time before this particular visit), the errors work themselves out nicely during the discussion of the problem.
So, for this particular problem, I got the following 4 answers:
I wish I had thought to take a photo of the board after we were finished (need to get better at documenting things for the blog!). First, I had a few students who defended the correct answer of 6 with some good strategies – adding on, counting back, making 44 into 45 etc. If I had taken a picture, you could note my nice use of number lines and whatever else I did to record student thinking…
But what I really want to discuss is the student who wanted to defend the answer of 14.
She said something like:
“I did the 5 minus the 4 to get 1 and the 4 minus the 0 to get 4. The answer is 14.”
This is not really earth-shattering… probably the most common mistake made in subtraction by Grade 4 students. Here is the interesting thing: this student had just listened to 3 or 4 of her peers defend (very clearly) their answer of 6 with very good strategies (and I’m quite sure she was listening). This is the first time I have had a student sincerely defend a mistake after multiple other students have made their case for the right answer – she had no recognition that her answer might not be correct. In hindsight, I can think of quite a few good ways to respond, but in the moment, I was caught off-guard. I wish I could tell you that I referred her back to the original problem to see if her answer made sense… or that I asked a classmate to respond to her thinking… or that I asked her to explain why what she did made sense to her. But… I just told her that you couldn’t flip the numbers around and subtract from bottom to top. Sigh. Fail.
Even in the moment, I knew that my response was woefully inadequate… I could tell from the look on her face that I had done nothing to convince her. I think she believed me that her answer was wrong, but she had gained no understanding to move her thinking forward. Other students had not learned anything useful from our exchange. And, possibly (hopefully not!), the experience has discouraged her from taking another risk to share her thinking.
So, what have I taken away from this experience? I went back to Making Number Talks Matter and reminded myself of some guiding principles…
Through our questions we seek to understand students’ thinking: It is not my role to be the judge of student answers, or even to correct mistakes. It is my role to try to understand why students are thinking the way they are. I need to focus my responses on questioning with the genuine desire to understand student thinking.
One of our most important goals is to help students develop social and mathematical agency: This exchange would have been a great opportunity to encourage students to respond to each other. By “explaining” the right answer, I removed the opportunity for students to be the thinkers and brought the responsibility for “correct mathematical knowledge” back to the teacher. My new #1 goal for number talks: stop talking so much and LISTEN.
Confusion and struggle are natural, necessary, and even desirable parts of learning mathematics:In hindsight, it is really interesting how uncomfortable I felt dealing with this mistake… as teachers, it is very hard to let go of our instincts to help our students through their struggles. I am totally on board with the IDEA of stepping back and letting my students wrestle with mistakes, but in the moment, it is still a challenge to stop the “traditional teacher” who hides out in the back of my brain.
I am thankful that teaching is such an interesting job – regardless of how much experience we have, there is always more to learn.
I am going to better prepare for my number talks… I have been lazy about anticipating student responses. For our last workshop, we prepared a “cheat sheet” of phrases and sentence stems, and I have printed a copy to refer to.
I am giving myself some grace… making mistakes is the best way to learn, even for teachers!
And I found this lovely quote from Ruth Parker to help me remember why I am so excited about doing Number Talks in the first place:
I’ve come to believe that my job is not to teach my students to see what I see. My job is to teach them to see.
So… who else wants to ‘fess up? What surprises have you been faced with during a Number Talk?
This fall, we (the two intermediate math liaisons in my district) have been planning a book study for the book “Making Number Talks Matter” by Cathy Humphreys and Ruth Parker. Our 18 participants teach Grades 4-7 and come from 16 different schools across our district (our district has 31 elementary schools). We will be meeting every second Tuesday until the beginning of December to work our way through the book. We will be talking about number talks, strategies for mental math and doing some planning and practicing of the Number Talks routine. Participation in this book club is totally voluntary and we know how difficult it is as teachers to squeeze in after-school commitments and still have everything ready in the classroom – our book club meetings run from 3:30 – 4:30 and we have committed to getting everyone out of there on time.
Yesterday was our first meeting – we had a few people who had to miss the first meeting because of parent-teacher interviews and staff meetings, so I will do my best to recap our discussion and learning!
We had a few goals for our first meeting –
Understand why we do number talks
Identify the procedures and setup necessary to get Number Talks started
Discuss the underlying values that the routine of Number Talks is based on
Create a plan for doing a dot talk in the classroom before the next meeting
We first showed the teachers a clip from the DVD that comes with Sherry Parrish’s Number Talks book. We asked the teachers to ignore the mathematical strategies (for now) and just to focus on the routine – what is the teacher doing? What are the students doing? What logistics do you notice? (View this YouTube video from 44:50 to 51:30 – this isn’t the exact same clip we watched, but close enough).
Afterwards, we asked each group of teachers to fill in a chart with their ideas from watching the video and from doing the pre-reading. Here are the finished ideas:
We then had teachers do a “gallery walk” to look at all the ideas.
Next, we had intended to pull up the new BC Curriculum website to show all the places that Number Talks fit in both the content elaborations and in the Curricular Competencies from Grades 4-7, but the website was down (*deep breath*), so we skipped this portion. We are planning to do a full workshop on our district’s next ProD day on Number Talks and the Curricular Competencies, so we will have a chance to dive into this further on that day (if you are from SD57 and reading this, you can register on PD Reg for this session!).
From here, we provided groups with some discussion questions from the chapters they read and gave them a few minutes to talk/discuss and plan:
What strikes you as most useful/valuable/exciting about the Number Talks routine?
What parts of the routine are of concern? What do you think will be most difficult for you as the teacher/facilitator?
What norms and structures do you need to have in place to be successful with Number Talks?
What Guiding Principles (from Chapter 3) resonate with you?
Which ones make you feel uncomfortable/concerned?
We provided groups with a blank template to record some guiding principles/norms for Number Talks that they thought they might like to use in their classrooms. There was so much good discussion during this portion of our meeting – I feel so lucky that I get to facilitate and work with groups of teachers on things like this – what a thoughtful group of people! During these conversations, teachers discussed the importance of “wait time” and how difficult that can be, they talked about the difficulties of facilitating if we ourselves are unsure about some of the strategies (hopefully we can clear some of these feelings up in future meetings), they talked about the importance of students listening to one another and how this routine can connect across many math content areas.
Finally, I did a demonstration “dot talk” so that teachers could see what a dot talk would look like in action. I used the same dot pattern from the Chapter explanation and showed teachers briefly how I set up a number talk to get started.
For our next meeting in two weeks, we have asked teachers to try a dot talk (or several) in their classroom, and read Prelude to the Operations, Chapter 4 and Chapter 6 – we are going to dive into addition and subtraction number talks next.
These are the Guiding Principles that I use when I start Number Talks in a classroom – they are adapted from Making Number Talks Matter.
Here is a planning page that Dorianna and I made for a workshop we did last year on Number Talks (also adapted from Making Number Talks Matter).
I borrowed and adapted several ideas for this session from this blog – I am so thankful for teachers who share their ideas and work so graciously online!
This is a great summary of Number Talks if anyone is looking for more information.
I had good intentions to blog this summer, but rest and family time ended up taking priority. It’s October already!?! So I am trying (again) to be committed to this blogging thing.
One thing I did manage to do this summer was some reading. Professional reading is a bit of a wormhole. One book leads to the next, which leads to the next and I always seem to have about 5 waiting for me to get to them. This is what I managed to read this summer:
Essentialism: the Disciplined Pursuit of Less by Greg McKeown
This is the best productivity book I have ever encountered. It is minimalism, but for your time instead of for your stuff. I loved everything about this book, but a few things really resonated with me.
It is impossible to do it all, so set some selective criteria that help to outline what you really want to accomplish and then STICK TO IT!
Create a buffer by adding 50% to your estimate of how long it will take to accomplish things. I am a chronic under-estimator of how much time things will take and often have to pull things together at the last minute. I’m sure I would experience more EASE in my life if I consciously added in a buffer.
Set aside professional time to think and read – this is really hard to do as a teacher – there are so many demands on our time. But some of the most creative insights and solutions to problems come when I give my mind time and space to think. I want to be intentional about adding this kind of time to my workweek this year.
The Innovator’s Mindset by George Couros
This book is related so closely to the shift our province is currently making in our curriculum – it is so much more important to teach our students HOW to think and learn rather than worrying about WHAT they are learning. There is currently a MOOC going on as a book study with this book that I was trying to keep up with, but… I am about 2 weeks behind (see the comment about the buffer above). Luckily, the Live chats are being archived, so I can follow at my own pace. (#IMMOOC if you are interested).
What’s Math Got to Do With It by Jo Boaler
I applied to read this book and write a review for the BC Association of Math Teachers book club series. Jo Boaler’s books are so inspiring and her YouCubed website is full of great resources. You can read my full review of the book here when it gets posted.
Classroom Chef by John Stevens and Matt Vaudry
I enjoyed the creativity of the lesson ideas and tips around crafting lessons. I think many teachers feel anxious about straying too far from “predictable” in Math class and this book reminds us that there are rewards for doing so. I actually (for the first time ever) thought it might be fun to teach a Grade 8 or 9 class. Luckily, that feeling has passed quickly 🙂
Teach Like a Pirate by Dave Burgess
I am a couple of years behind the bandwagon on this one, but I have had it signed out of the district library a few times and have never made time to get through it. I am glad I finally read it – there are so many great ideas for making lessons interesting and motivating for students. This book was a good reminder of why I became a teacher in the first place. A very motivating read!
And… that brings me to the present…
Recently, I have been very intrigued by the idea of thinking routines and instructional routines that support deep thinking. Making Thinking Visible caught my eye on Amazon and after ordering it, I am seeing references to it everywhere – a good sign.
I started Mathematical Mindsets in the spring and got about halfway through it – I really wanted to read it slowly and carefully because there is so much to think about. I am looking forward to digging back in this fall. This is the book I am considering for my next teacher book club – depending on how successful our Number Talks book club is this fall (more on this next week!).
So, that’s my professional reading life over the last while… does anyone have any good suggestions for what to read next?
This is my (admittedly ambitious) summer professional reading list. I have just finished the first book (reflection coming soon) and Making Number Talks Matter is a re-read to plan a book study in my district next year. So – hopefully it will be do-able!
What professional books are you planning to read this summer?
As the end of the year approached, I was invited to bring some open-ended problem solving activities to a few classrooms. A colleague that I met in a Math book study earlier this year had introduced me to 3 Act Math tasks and I thought it would be a perfect opportunity to try some.
I visited two Grade 4/5 classes and used Graham Fletcher’s Array-bow of Colours task. One of my favourite things about my job as a math coach is that I frequently get the chance to teach the same lesson twice in a short time span – such a great opportunity for reflecting and thinking about my practice.
I began in both classes by asking students to Notice and Wonder about the Act 1 video. Here were some thoughts from the first class I was in:
We focussed on the “how many skittles” question for this activity, but I love the variety of questions they came up with and I love how many of them were actually measureable questions. I was pleased that someone came up with the question: “What if not every small pack has the same number of skittles?” I actually brought packages of skittles for the kids to work with (everything is more fun with candy) and I had debated how to handle this question, so I was happy that one of the students thought of it before we got started. I was also so happy that it was a student who is typically not very engaged, and who doesn’t consider herself to be “good” at math. It was a nice opportunity to really value her thinking!
We did some estimating and, as usual, had a HUGE variety of thoughts here. You can see the range of the estimates in the first picture above – the lowest estimate in the class was 50 and the highest was 1050.
When we got to Act 2, the students knew they wanted to know the number of packages and number of skittles/package. This class (and school) has a focus on multiplication as their essential skill for this year, so pretty much every group (partners) knew that they were going to multiply and no one had a hard time getting started. Most groups quickly found the answer to 58 x 14 was 812. The interesting challenge for this class was how they were going to address the variation in the packages. As soon as students started opening their skittles, it became apparent that not every package was the same… 14 was the low end of the scale. The highest number of skittles per package was 19 and most kids had something in between. To account for this, some students added a constant at the end to account for there being “some” more skittles (ie. 812 + 50). Some students changed their multiplication question – 58 x 16 (because 16 is between 14 and 19). Some did two calculations – 58 x 14 and 58 x 19 and then picked a number in between. This gave us opportunities for some rich discussions in Act 3.
I did the same activity with another 4/5 class the following day and noticed some interesting differences. The second class was in an inner city school and they have not had the focus on multiplication as an essential skill. They generally struggled a lot more with this activity, but still had some good observations and ideas. Their estimates were wildly off – the range for the class was between 100 and 400. Overall, they knew they were going to multiply, but didn’t have the skills to do the 2×2 digit multiplication. They were much more confused about the variation in number of skittles/package. In hindsight, knowing that they were less comfortable with multiplication, I might have waited to hand out the skittles packages until the end and then we could have had a whole-class discussion about what to do with the variation in package size. I think this would have filtered out some of their confusion during act 3 – some students were really quite paralyzed by the variation in packages. I still think this activity was valuable for them.
One of the things I like about this particular task is the photo provided for Act 3. This gave us a neat little opportunity to talk about efficient ways of counting/grouping – it made for a good visual number talk at the end of the task.
I also recently visited a Grade 6/7 class and I used Dan Meyer’s “Super-Bear” task. The kids were really engaged (which is quite impressive for this particular class) and all groups ended up with a workable strategy. If I were going to do this again, I would encourage the use of calculators – most lots of groups just got bogged down in a problem that required long division with decimals (this is understandable… I’m pretty sure I would get bogged down tackling a long division with decimals problem too). I think the value in this activity is in the problem-solving nature and not in the calculation piece.
I am eager to try out some more 3 Act Math.
I love that they are such an engaging way to have kids problem solve.
I enjoy giving the responsibility for all the thinking to the students… these tasks really make me feel like I am a facilitator rather than a teacher.
I am really going to encourage teachers and schools next year to pick a topic or idea to really focus on in their Math instruction. I have been so impressed by the school that I have been working in that has really narrowed down their approach and really worked collaboratively to ensure that students progress.
There are so many of these tasks available online and for such a wide range of grades and topics -I am so grateful to all the teachers who have spent time developing these and sharing so generously online!
If you are new to 3 Act Math tasks, I would highly recommend this video by Dan Meyer in which he demonstrates the implementation.
Happy summer to any other teachers out there who had their last day today!!